# Properties

 Label 1617.2.a.x Level $1617$ Weight $2$ Character orbit 1617.a Self dual yes Analytic conductor $12.912$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(1,1617)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$12.9118100068$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 - q^3 + (b2 + b1 + 1) * q^4 + (b2 - b1) * q^5 - b1 * q^6 + (b3 + 2*b2 + b1 + 4) * q^8 + q^9 $$q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{8} + q^{9} + (\beta_{3} - 2) q^{10} - q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{12} - \beta_{3} q^{13} + ( - \beta_{2} + \beta_1) q^{15} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 3) q^{16} + (2 \beta_{2} + \beta_1) q^{17} + \beta_1 q^{18} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{19} + ( - \beta_{2} - \beta_1) q^{20} - \beta_1 q^{22} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{23} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{24} + ( - 2 \beta_{3} - 2 \beta_{2} - 1) q^{25} + ( - \beta_{2} + \beta_1) q^{26} - q^{27} + ( - \beta_{3} - \beta_1 + 2) q^{29} + ( - \beta_{3} + 2) q^{30} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{31} + (4 \beta_{2} + 5 \beta_1 + 6) q^{32} + q^{33} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 5) q^{34} + (\beta_{2} + \beta_1 + 1) q^{36} + ( - \beta_{2} - \beta_1 - 1) q^{37} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{38} + \beta_{3} q^{39} + ( - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{40} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + ( - 2 \beta_{2} + 3 \beta_1 + 2) q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{44} + (\beta_{2} - \beta_1) q^{45} + ( - 4 \beta_{2} + \beta_1 - 6) q^{46} + ( - \beta_{2} - \beta_1 - 3) q^{47} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{48}+ \cdots - q^{99}+O(q^{100})$$ q + b1 * q^2 - q^3 + (b2 + b1 + 1) * q^4 + (b2 - b1) * q^5 - b1 * q^6 + (b3 + 2*b2 + b1 + 4) * q^8 + q^9 + (b3 - 2) * q^10 - q^11 + (-b2 - b1 - 1) * q^12 - b3 * q^13 + (-b2 + b1) * q^15 + (2*b3 + 2*b2 + 4*b1 + 3) * q^16 + (2*b2 + b1) * q^17 + b1 * q^18 + (b3 + 2*b2 - b1 + 2) * q^19 + (-b2 - b1) * q^20 - b1 * q^22 + (-2*b3 - 2*b1 + 1) * q^23 + (-b3 - 2*b2 - b1 - 4) * q^24 + (-2*b3 - 2*b2 - 1) * q^25 + (-b2 + b1) * q^26 - q^27 + (-b3 - b1 + 2) * q^29 + (-b3 + 2) * q^30 + (-2*b3 + b2 + b1 + 2) * q^31 + (4*b2 + 5*b1 + 6) * q^32 + q^33 + (2*b3 + 3*b2 + 3*b1 + 5) * q^34 + (b2 + b1 + 1) * q^36 + (-b2 - b1 - 1) * q^37 + (2*b3 + 2*b2 + 2*b1 - 1) * q^38 + b3 * q^39 + (-3*b3 - 2*b2 - 2*b1) * q^40 + (b3 - 2*b2 + 2*b1 - 2) * q^41 + (-2*b2 + 3*b1 + 2) * q^43 + (-b2 - b1 - 1) * q^44 + (b2 - b1) * q^45 + (-4*b2 + b1 - 6) * q^46 + (-b2 - b1 - 3) * q^47 + (-2*b3 - 2*b2 - 4*b1 - 3) * q^48 + (-2*b3 - 4*b2 - b1 - 2) * q^50 + (-2*b2 - b1) * q^51 + (b3 + 2) * q^52 + (4*b3 + b2 + 3*b1 - 2) * q^53 - b1 * q^54 + (-b2 + b1) * q^55 + (-b3 - 2*b2 + b1 - 2) * q^57 + (-2*b2 + 2*b1 - 3) * q^58 + (-2*b3 + 2*b1 - 5) * q^59 + (b2 + b1) * q^60 + (-b3 - 2*b1) * q^61 + (b3 + 6*b1 + 4) * q^62 + (5*b2 + 7*b1 + 13) * q^64 + (2*b3 + 2*b2 - 2*b1 + 2) * q^65 + b1 * q^66 + (2*b3 + b2 - 3*b1 + 10) * q^67 + (3*b3 + 4*b2 + 7*b1 + 12) * q^68 + (2*b3 + 2*b1 - 1) * q^69 + (2*b3 + b2 - b1 + 5) * q^71 + (b3 + 2*b2 + b1 + 4) * q^72 + (b3 - 6*b2 - 4) * q^73 + (-b3 - 2*b2 - 3*b1 - 4) * q^74 + (2*b3 + 2*b2 + 1) * q^75 + (2*b2 + 3*b1 + 4) * q^76 + (b2 - b1) * q^78 + (-3*b3 + 2*b2 - 4*b1 + 2) * q^79 + (-2*b3 - 5*b2 + b1 - 8) * q^80 + q^81 + (-2*b3 + b2 - 3*b1 + 4) * q^82 + (-4*b3 - 2*b1 + 2) * q^83 + (-b3 - 4*b2 + 2) * q^85 + (-2*b3 + b2 + 3*b1 + 7) * q^86 + (b3 + b1 - 2) * q^87 + (-b3 - 2*b2 - b1 - 4) * q^88 + (-6*b2 + 2*b1 - 6) * q^89 + (b3 - 2) * q^90 + (-3*b2 - 5*b1 - 3) * q^92 + (2*b3 - b2 - b1 - 2) * q^93 + (-b3 - 2*b2 - 5*b1 - 4) * q^94 + (-5*b3 - 4*b2 + 4) * q^95 + (-4*b2 - 5*b1 - 6) * q^96 + (-2*b3 + b2 - b1 + 11) * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9 $$4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9} - 10 q^{10} - 4 q^{11} - 4 q^{12} + 2 q^{13} + 4 q^{15} + 12 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{22} + 4 q^{23} - 12 q^{24} + 4 q^{25} + 4 q^{26} - 4 q^{27} + 8 q^{29} + 10 q^{30} + 12 q^{31} + 26 q^{32} + 4 q^{33} + 16 q^{34} + 4 q^{36} - 4 q^{37} - 8 q^{38} - 2 q^{39} + 6 q^{40} - 2 q^{41} + 18 q^{43} - 4 q^{44} - 4 q^{45} - 14 q^{46} - 12 q^{47} - 12 q^{48} + 2 q^{50} + 2 q^{51} + 6 q^{52} - 12 q^{53} - 2 q^{54} + 4 q^{55} - 4 q^{58} - 12 q^{59} - 2 q^{61} + 26 q^{62} + 56 q^{64} - 4 q^{65} + 2 q^{66} + 28 q^{67} + 48 q^{68} - 4 q^{69} + 12 q^{71} + 12 q^{72} - 6 q^{73} - 16 q^{74} - 4 q^{75} + 18 q^{76} - 4 q^{78} + 2 q^{79} - 16 q^{80} + 4 q^{81} + 12 q^{82} + 12 q^{83} + 18 q^{85} + 36 q^{86} - 8 q^{87} - 12 q^{88} - 8 q^{89} - 10 q^{90} - 16 q^{92} - 12 q^{93} - 20 q^{94} + 34 q^{95} - 26 q^{96} + 44 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 + 12 * q^8 + 4 * q^9 - 10 * q^10 - 4 * q^11 - 4 * q^12 + 2 * q^13 + 4 * q^15 + 12 * q^16 - 2 * q^17 + 2 * q^18 - 2 * q^22 + 4 * q^23 - 12 * q^24 + 4 * q^25 + 4 * q^26 - 4 * q^27 + 8 * q^29 + 10 * q^30 + 12 * q^31 + 26 * q^32 + 4 * q^33 + 16 * q^34 + 4 * q^36 - 4 * q^37 - 8 * q^38 - 2 * q^39 + 6 * q^40 - 2 * q^41 + 18 * q^43 - 4 * q^44 - 4 * q^45 - 14 * q^46 - 12 * q^47 - 12 * q^48 + 2 * q^50 + 2 * q^51 + 6 * q^52 - 12 * q^53 - 2 * q^54 + 4 * q^55 - 4 * q^58 - 12 * q^59 - 2 * q^61 + 26 * q^62 + 56 * q^64 - 4 * q^65 + 2 * q^66 + 28 * q^67 + 48 * q^68 - 4 * q^69 + 12 * q^71 + 12 * q^72 - 6 * q^73 - 16 * q^74 - 4 * q^75 + 18 * q^76 - 4 * q^78 + 2 * q^79 - 16 * q^80 + 4 * q^81 + 12 * q^82 + 12 * q^83 + 18 * q^85 + 36 * q^86 - 8 * q^87 - 12 * q^88 - 8 * q^89 - 10 * q^90 - 16 * q^92 - 12 * q^93 - 20 * q^94 + 34 * q^95 - 26 * q^96 + 44 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 2$$ v^3 - 2*v^2 - 3*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 5\beta _1 + 4$$ b3 + 2*b2 + 5*b1 + 4

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.51658 −0.552409 1.28734 2.78165
−1.51658 −1.00000 0.300014 2.33317 1.51658 0 2.57816 1.00000 −3.53844
1.2 −0.552409 −1.00000 −1.69484 −1.59002 0.552409 0 2.04107 1.00000 0.878345
1.3 1.28734 −1.00000 −0.342766 −3.91744 −1.28734 0 −3.01593 1.00000 −5.04306
1.4 2.78165 −1.00000 5.73760 −0.825711 −2.78165 0 10.3967 1.00000 −2.29684
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$-1$$
$$11$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1617.2.a.x 4
3.b odd 2 1 4851.2.a.bu 4
7.b odd 2 1 1617.2.a.z 4
7.d odd 6 2 231.2.i.e 8
21.c even 2 1 4851.2.a.bt 4
21.g even 6 2 693.2.i.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.e 8 7.d odd 6 2
693.2.i.i 8 21.g even 6 2
1617.2.a.x 4 1.a even 1 1 trivial
1617.2.a.z 4 7.b odd 2 1
4851.2.a.bt 4 21.c even 2 1
4851.2.a.bu 4 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 4T_{2} + 3$$ T2^4 - 2*T2^3 - 4*T2^2 + 4*T2 + 3 $$T_{5}^{4} + 4T_{5}^{3} - 4T_{5}^{2} - 20T_{5} - 12$$ T5^4 + 4*T5^3 - 4*T5^2 - 20*T5 - 12 $$T_{13}^{4} - 2T_{13}^{3} - 8T_{13}^{2} + 16T_{13} - 4$$ T13^4 - 2*T13^3 - 8*T13^2 + 16*T13 - 4 $$T_{17}^{4} + 2T_{17}^{3} - 40T_{17}^{2} - 124T_{17} + 15$$ T17^4 + 2*T17^3 - 40*T17^2 - 124*T17 + 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 3$$
$3$ $$(T + 1)^{4}$$
$5$ $$T^{4} + 4 T^{3} + \cdots - 12$$
$7$ $$T^{4}$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} - 2 T^{3} + \cdots - 4$$
$17$ $$T^{4} + 2 T^{3} + \cdots + 15$$
$19$ $$T^{4} - 40 T^{2} + \cdots - 89$$
$23$ $$T^{4} - 4 T^{3} + \cdots + 465$$
$29$ $$T^{4} - 8 T^{3} + \cdots + 3$$
$31$ $$T^{4} - 12 T^{3} + \cdots - 1396$$
$37$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$41$ $$T^{4} + 2 T^{3} + \cdots + 60$$
$43$ $$T^{4} - 18 T^{3} + \cdots - 1385$$
$47$ $$T^{4} + 12 T^{3} + \cdots + 9$$
$53$ $$T^{4} + 12 T^{3} + \cdots + 6252$$
$59$ $$T^{4} + 12 T^{3} + \cdots + 249$$
$61$ $$T^{4} + 2 T^{3} + \cdots + 20$$
$67$ $$T^{4} - 28 T^{3} + \cdots - 1460$$
$71$ $$T^{4} - 12 T^{3} + \cdots - 699$$
$73$ $$T^{4} + 6 T^{3} + \cdots + 17156$$
$79$ $$T^{4} - 2 T^{3} + \cdots + 388$$
$83$ $$T^{4} - 12 T^{3} + \cdots + 2592$$
$89$ $$T^{4} + 8 T^{3} + \cdots + 12048$$
$97$ $$T^{4} - 44 T^{3} + \cdots + 8501$$